Showing papers on "Voltage graph published in 1985"

Applications of graph theory in chemistry

Abstract: Graph theoretical (GT) applications in chemistry underwent a dramatic revival lately. Constitutional (molecular) graphs have points (vertices) representing atoms and lines (edges) symbolizing malent bonds. This review deals with definition. enumeration. and systematic coding or nomenclature of constitutional or steric isomers, valence isomers (especially of annulenes). and condensed polycyclic aromatic hydrocarbons. A few key applications of graph theory in theoretical chemistry are pointed out. The complete set of all poasible monocyclic aromatic and heteroaromatic compounds may be explored by a mmbination of Pauli's principle, P6lya's theorem. and electronegativities. Topological indica and some of their applications are reviewed. Reaction graphs and synthon graphs differ from constitutional graphs i n their meaning of vertices and edges and find other kinds of chemical applications. This paper ends with a review of the use of GT applications for chemical nomenclature (nodal nomenclature and related areas), coding. and information processing/storage/retrieval

Protocol validation by fair progress state exploration

Abstract: Consider a network of two communicating finite state machines that exchange messages over two one-directional, unbounded, FIFO channels. The fair reachability graph of such a network is a directed graph whose vertices correspond to global states (of the network) that are reachable by forcing the two machines in the network to progress in equal speeds. It is shown earlier that if the fair reachability graph of a given network is finite, then it can be used to decide whether the communication of this network is free from deadlocks and unspecified receptions. In this paper, we complement this result by showing that if the fair reachability graph of a given network is finite, then it can be used to decide wether the communication of this network is bounded. Moreover, if the communication is found to be bounded, then the finite fair reachability graph can be also used to compute the smallest possible capacities for the two channels in the network.

Developments in the theory of graph spectra

Abstract: Some of the main directions and newer results in the theory of graph spectra are reviewed. Some areas discussed are (i) the application of root systems to the theory of graph spectra, (ii) results concerning spectral characterizations of graphs with least eigenvalue-2, (iii) a description of some new graph invariants based on the eigenvectors of the adjacency matrix, along with (iv) the algebraic solution of the Shannon capacity problem, (v) results on spectra of random graphs, and (vi) a review of some graph polynomials related to the characteristic polynomial. Finally, (vii) a discussion of recent results concerning the spectra of infinite graphs is given.

Lower-bounds on the connectivities of a graph

Abstract: This article presents a study of the connectivities of a graph as a function of other graph parameters such as the number of vertices, the maximum degree, and the diameter. As a result, lower-bounds on the connectivities of a graph as a function of these parameters are computed. These bounds could serve as sufficient conditions for a graph to be h-edge-connected or k-connected. Consequently, the connectivity characteristics of many of the densest known graphs are determined.

Tree-Based Graph Algorithms for Some Parallel Computers.

Abstract: This paper gives several optimal mesh computer, VLSI, and pyramid computer algorithms for determining properties of an arbitrary undirected graph, where the graph is given as an unordered collection of edges. The algorithms first find spanning trees and then use them to determine properties of the graph. By using edges, instead of requiring an entire adjacency matrix, these algorithms use only time on a 2-dimensional mesh, instead of the time required with matrix input. Further, the edge-based algorithms extend naturally to meshes of arbitrary dimension ,fi nishing in time. All of the times are optimal, and the algorithms extend to VLSI and pyramid models.

The splitting number of the complete graph

Abstract: If a given graphG can be obtained bys vertex identifications from a suitable planar graph ands is the minimum number for which this is possible thens is called the splitting number ofG. Here a formula for the splitting number of the complete graph is derived.

Amalgamation of Graph Transformations with Applications to Synchronization

Abstract: In the present paper we generalize the well-known PARALLELISM THEOREM for graph derivations to the AMALGAMATION THEOREM. In this theorem the assumption of ‘parallel independence’ is dropped. For each pair of productions together with a relational production (allowing productions to be associated with each other) we construct a single ‘amalgamated’ production. The AMALGAMATION THEOREM states that graph derivations which respect the given associations can be amalgamated to a single derivation via the ‘amalgamated’ production.

On self-complementation

Abstract: We prove that, with very few exceptions, every graph of order n, n - 0, 1(mod 4) and size at most n - 1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs.

The gradient of a graph

Abstract: In this article I introduce a dynamic interpretation of the gradient of a graph which leads naturally into the notion of differentiation.

An approximate formula for counting trees in a graph

Abstract: In the paper, an approximate formula for counting trees in a graph is presented. The formula estimates the change in the number of trees of a reference graph after adding or removing a certain number of lines to obtain a graph with the same number of lines as the graph considered. It is assumed that the reference graph and the graph analyzed have the same number of points. It is also assumed that the number of trees of a reference graph changes with a change in the number of lines identically as in a uniform graph. Special classes of reference graphs are discussed and formulas for counting trees are given.

Any four independent edges of a 4-connected graph are contained in a circuit

Abstract: Conjecture. Suppose G is a k-connected graph (k=>2), el, e~ . . . . , ekEE(G) are independent edges, and if k is odd then G {el, e2, ..., ek} is connected. Then G contains a circuit using all the edges e~, e~ . . . . , e a. This conjecture is proved for k=3 by Lov~sz [3; 6. w 67]. In general, R. Hiiggkvist and C. Thomassen [1] proved a slightly weaker statement that the same conclusion follows if G is (k+ 1)-connected. Now we prove that the conjecture of Lov~isz holds for k=4.

O(1) Parallel Time Incremental Graph Algorithms

Abstract: Fast parallel algorithms are presented for updating minimum spanning trees, connected components and bridges of an undirected graph when a minor change is made to the graph such as addition or deletion of a vertex or an edge. The machine model used is a parallel random access machine that allows simultaneous reads as well as simultaneous writes into the same memory location. In the latter case one processor succeeds but we do not know which. The algorithms described in this paper require O(1) time and are efficient when compared to previously known O(logn) time algorithms for initial computation of the above mentioned graph properties on this model. An important feature of our algorithms is their versatility, that is, they can be adapted to run efficiently on all variations of this model with very little modification.

A basis for the Cycle Space of a 3-Connected Graph

Abstract: Let G be a 3-connected non-hamiltonian graph with minimum degree d. We prove that the cycles of length at least 2d-l generate the cycle space of G.

A Method of Rapid Markov Reliability Calculation

Abstract: Our reliability calculation method for a Markov state-transition graph enables a rapid computation by finding and cutting (removing) non-effective edges (NEEs). An NEE is an edge in a Markov state transition graph, the cut of which has little effect on the reliability calculation. NEEs can be found only by checking in a small graph, given the assumption that component failure rate is far smaller than component repair rate. NEEs are found and cut until the Markov graph is separated into two subgraphs. One subgraph is usually very small compared with the original, and the reliability can be approximately calculated on this small subgraph of the Markov graph. Proof and numerical examples are presented.

A decomposition of a graph into dense subgraphs

Abstract: A density of a graph is defined as the maximum number of disjoint spanning trees of the graph. Using this density an algorithm to find all the subgraphs of a given density of a graph is developed. The decomposition is unique when a density is given. The algorithm is efficient.

Homology classes which are represented by graph links

Abstract: We give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link. The purpose of this paper is to give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link. For this, we define the Jaco-Shalen-Johannson complex of a graph manifold (?1), using the so-called torus decomposition theorem due to Jaco-Shalen and Johannson (see also Waldhausen [2]), and the main result is stated as follows (?2): THEOREM. Let M be a graph manifold prime to S1 X S2 and p: M -T WM the natural map to the Jaco-Shalen-Johannson complex of M. Then an element a of H1(M; Z) can be represented by a graph link if and only if p*o(a) = 0 in Hl(WM; Z). If the ambient manifold M is not prime to S1 X S2, the homotopy class of the map p: M -T WM is not unique in general, and this makes the statement complicated (?3). The proof of these results is based on the study of global graph links in [3]. 1. Preliminaries. Throughout this paper, manifolds are compact, oriented, of dimension three and with toral boundary, and links are oriented. A manifold M is a graph manifold if there is a family of disjointly embedded tori in M such that each connected component of the manifold obtained by cutting M along these tori is the total space of an S1-bundle over a surface. A link is called a graph link if its exterior is a graph manifold. We say that a manifold is prime to S1 X S2 if its prime decomposition does not contain S1 X S2. See Jaco [1] and Yano [3] for other terminology. To state our result, we need the following theorem due to Jaco-Shalen and Johannson (see Jaco [1] and also Waldhausen [2]). THEOREM 1.1 (JACO SHALEN, JOHANNSON). Let M be a Haken manifold which is either closed or with incompressible boundary. Then there exists a unique Seifert submanifold E c M up to ambient isotopy such that (1) E is maximal, and (2) if E' is a Seifert manifold pair distinct from (S3, 0), (S2 X SI, 0), (D2 X I, aD 2 X I) or (S1 X D2, 0), then every nondegenerate map f: E' -> (M, aM) is homotopic to fo such that fo(E') c E. The E above is called the characteristic Seifert manifold of M. Received by the editors September 19, 1983 and, in revised form, March 14, 1984. 1980 Mathematics Subject Classification. Primary 57M99, 57N10.

The erdös-dushnik-miller theorem for topological graphs and orders

Abstract: A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdos, Dushnik and Miller, for any infinite set L⊑V, there is a subset L′⊑L of the same oardinality |L′| = |L| such that the restriction G ↾ L′ is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset A⊑V does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L′ as before, but if cf (|L|) ω, there is a subset L′⊑L of size |L′|=|L| such that G↾L′ is complete. The condition cf (|L|)>ω is needed.